Haven't listened to the NPR story, but I'm somewhat skeptical.
Plant competition is well-established. There is also a decent body of work demonstrating several examples of plants exuding chemicals that place other plant species at a disadvantage. The examples I can think of were smaller plants that put chemicals into the soil near their roots.
The thing is, putting chemicals in discarded leaves seems like a rather inefficient mechanism for poisoning the soil - the leaves would spread out all over the place, and most of the chemicals would probably be broken down by microorganisms before they could be taken up by competing trees' roots.
It might work (for a sufficiently long-lasting poison) if you assume that a given species of trees already has a dominant position and can effectively blanket the downwind area, but in that scenario the "shirking" problem would almost certainly outweigh the evolutionary benefit. That is, any "lazy" mutant that spared itself the effort of making all those poisons would be able to use that energy for growth and reproduction, while its seeds would still benefit from the leaf-poisons being produced by its neighbors. Nothing would stop the "lazy" trees from eventually outnumbering the poisoners. (Exception: trees that reproduce primarily by cloning wouldn't have a shirking problem. However, they also wouldn't get any benefit from scattering their leaves to the wind - I don't know a tree that disperses clonal seeds/spores on the wind.)
Somebody said they thought trees put chemicals in their leaves to get rid of the harmful chemicals. The problem with that theory is, the chemicals in plant leaves are, almost exclusively, *synthesized* by the plant itself. Plants have lots of chemicals in their leaves, primarily because the presence of strange chemicals has the evolutionary advantage of making the leaf less attractive as a foodstuff (mostly for insects, but also for larger browsers like mammals). That's how we got tobacco, and also milkweed. (In these two cases an insect has famously acquired resistance, but all other insects have been excluded.)
I'll have to listen to the original to find out what kind of chemicals we're talking about, but the theory seems pretty easily verifiable (though harder to falsify) - find a chemical X in a leaf of species A that inhibits the growth of species B, *more than it inhibits the growth of species A*. A "smoking gun" would be to find an unusual adaptation in A that makes it resistant to X. By itself, that's still not enough to eliminate other explanations for why A produces X (A might produce X as an insecticide and the inhibition of B might not have any real-world consequences because its leaves are scattered too far or X does not persist in the soil), but it would get my attention.
